Homepage International Economics

* International Economics*, Robert A. Mundell, New York: Macmillan,
1968, pp. 298-317.

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The stability analysis introduced by Hicks has been one of the most successful failures in economic theory. Originally developed to integrate statical and dynamical general equilibrium theory, it was used by Hicks ([24], p. 62) as a bridge between dynamics and comparative statics:

The laws of change of the price system, like the laws of change of individual demand, have to be derived from stability conditions. We first examine what conditions are necessary in order that a given equilibrium system should be stable; then we make an assumption of regularity; that positions in the neighbourhood of the equilibrium position will be stable also; and thence we deduce rules about the way in which the price-system will react to changes in tastes and resources.

But the stability conditions were not founded on an explicitly formulated dynamic system. Hicks had defined stability along conventional lines:

In order that equilibrium should be stable, it is necessary that a slight movement away from the equilibrium position should set up forces tending to restore equilibrium....

but concluded that stability requires that

. . . a rise in price makes supply greater than demand, a fall in price demand greater than supply.

This is indeed the stability condition corresponding to a dynamic system
in which excess demand causes a rise in price, but, as Samuelson pointed
out, the proposition is not explicitly derived as the condition of convergence
of such a dynamical system. The problem may be considered trivial in the
case of a single
market,^{3} but it raises difficulties in analysis
and interpretation in the case of multiple exchange (exchange of more than
two commodities), as Hicks foresaw ([24], p. 66):

What do we mean by stability in multiple exchange ? Clearly, as before, that a fall in the price ofXin terms of the standard commodity will make the demand forXgreater than the supply. But are we to suppose that it must have this effect (a) when the prices of other commodities are given, or (b) when other prices are adjusted so as to preserve equilibrium in the other markets?

To resolve the difficulty Hicks introduced his concepts of *perfect*
and *imperfect stability*. He noted first that it was necessary to
distinguish a series of conditions-that a rise in the price of *X* will
make supply greater than demand (1) all other prices being given, (2) allowing
for the price of *Y* being adjusted to maintain equilibrium in the
*Y* market, (3) allowing for the prices of *Y* and *Z* being
adjusted, and so on, until all prices have been adjusted. He then defined
as *imperfectly stable* a system in which a rise in price of a commodity
causes excess supply for the commodity after all repercussions are allowed
for, and as *perfectly stable* a system in which a rise in price causes
excess supply regardless of how many other prices are adjusted to attain
equilibrium values in their respective
markets.^{4}

Samuelson [88] criticized Hicks' concept of stability on the grounds, as
stated above, that "stability conditions are not deduced from a dynamic model,
except implicitly." Hicksian stability is not equivalent to "true" dynamic
stability, and the Hicks conditions are neither necessary nor sufficient
for the convergence of the dynamical system implicit in Hicks' dynamics,
that is, the dynamic system Samuelson postulated as the "natural extension"
of the Walrasian system. True dynamic stability requires that the roots of
the characteristic equation of the dynamic system have negative real parts,
and this requirement is not equivalent to the Hicks
conditions.^{5}

Samuelson's criticism undermined the logic of Hicks' method. But the conditions
of stability produced by that method retained an important place in the
literature. Samuelson had already observed that the Hicks conditions were
equivalent to the conditions of "true" dynamic stability in the symmetrical
case. Metzler showed that they were equivalent in the case of gross substitutes,
and also necessary (but not sufficient) for stability to be independent of
the speeds of adjustment (Metzler
[62]).^{6} Morishima proved that they were
equivalent for certain classes of complements. They are also sufficient
conditions for convergence of any *nonoscillating* system, since Hicksian
(perfect) stability implies the absence of positive real roots. They are,
moreover, conditions that, if not satisfied, yield anomalous comparative
statics results, and thus seem at least to be necessary conditions for useful
applications of the correspondence principle, at least in the context of
analysis of the Walrasian system. Thus, even though Hicks' method seems to
lack theoretical
justification,^{7} the Hicks *stability conditions*
produced by that method have proved exceedingly useful.

How can a wrong method yield useful results ? Leaving aside coincidence, the answer may lie in the two-way character of the correspondence principle. Samuelson [89] had observed that

not only can the investigation of the dynamic stability of a system yield fruitful theorems in statical analysis, but also known properties of a (comparative) statical system can be utilized to derive information concerning the dynamic properties of a system.

When Hicks is specifying the signs of changes in excess demands when a given
price is put above or below its equilibrium value, various subsets of other
prices remaining constant, he is at the same time implying specific comparative
statics results. Provided these results correspond to known properties of
a statical system the conditions implied should be related to stability
conditions if the *reciprocal* character of the correspondence principle
is valid.

Another reason why the *Hicks method* may appear more reasonable than
Samuelson's original criticism of it suggests is that our knowledge of the
precise laws governing dynamical systems is scanty. The empirical "output"
according to the methodology of the correspondence principle is a set of
comparative statics results, while the empirical "input" is (a) the nature
of the dynamic processes, and (b) the assumption of stability. Acceptance
of (b) is the essence of the correspondence principle, but how are we to
determine (a)?

Consider, for example, the following alternative expressions for dynamical systems:

To each of these systems there will correspond a different set of stability conditions. System (1) is a version of that used by Samuelson to prove that the Hicks conditions are neither necessary nor sufficient for stability. Yet, as he himself noted, more complete generalizations such as (2) and (3) can be developed with different consequences for comparative statics. There is, therefore, an element of arbitrariness in the specification of dynamic systems in the absence of empirical information and there may on these grounds be a pragmatic justification for Hicks' method of developing " stability conditions " that are "timeless." The Samuelson criterion is completely general and is an appropriate methodological approach, but for purposes of yielding practical results generality often implies emptiness.

The purpose of this chapter is to show that the Hicksian stability analysis
is a useful contribution to the integration of statical and dynamical theory.
First, we shall show that the perfect and imperfect stability conditions
do correspond to the dynamic stability conditions of *some* dynamic
processes, irrespective of the pattern of signs of the price-matrix. Second,
we shall argue that, despite their usefulness in the form Hicks presented
them, the perfect stability conditions are not completely general, since
they do not yield the information obtained by extending his method to the
commodity "adopted" as the standard commodity. Third, we shall show that
generalized conditions can be obtained by interpreting his device of holding
subsets of prices constant with respect to the standard commodity as an arbitrary
method of forming various composite commodity groupings. Further, we shall
show that dynamic systems that fail to satisfy the generalized conditions
will be unstable at *some* speeds of adjustment when a different commodity
is adopted as the standard commodity in the dynamic system. And finally,
we shall discuss the usefulness of the generalized Hicks conditions in devising
dynamical rules for the *hyperstability* of " policy
systems."^{8} The illustrative examples
are all taken from the theory of foreign exchange markets, but the results,
of course, apply to any generalized system.

Our first task is to show that the Hicks conditions do, in a sense, correspond
to the conditions of convergence of some dynamic systems. Let us take as
an example a problem in devaluation theory. We can describe a closed static
equilibrium system of *n* + 1 currencies with prices (exchange rates)
expressed in terms of currency 0, denoted by
*p*_{1}, . . .
*p*_{n}, as follows:

where *B*_{i} is the balance of payments of the
*i*th country. In equilibrium each
*B*_{i} = 0, while near the equilibrium we can
write the system (4) as follows:

after expanding *B*_{i} in a Taylor series and
omitting nonlinear terms.

Now let us suppose that the exchange rate of one country, say the *r*th
country, appreciates in proportion to its balance of payments surplus according
to the law

while all other exchange rates adjust instantaneously to equilibrium. The solution of the differential system (6) is

and *Delta*_{rr} is the cofactor (principal minor)
of the element in its *r*th row and *r*th column.

For the dynamic process implied in (7) to be stable it is necessary and
sufficient that *Delta* /
*Delta*_{rr} < 0. But this condition is precisely
(for the analogous problem in the Walrasian system) the Hicksian condition
of imperfect stability for the *r*th currency; and when the method is
applied to each currency (in succession, not simultaneously), we have the
complete Hicksian conditions of imperfect stability:

A similar analysis can help to show the usefulness of the Hicks conditions
of perfect stability. Suppose that one exchange rate, say
*p*_{i}, is held constant (relative to the
numéraire). This amounts to dropping the ith row and column from
*Delta*, so that if the original experiment were repeated, this time
with the ith exchange rate constant, we would get, instead of (7),

and the stability condition *Delta _{ii}* /

But does this dynamic process have any economic plausibility ? Are we not,
as Samuelson argued, allowing "arbitrary modification of the dynamical equations
of motion"? The answer is, in a sense, yes. But this can be the exact method
needed in the theory of policy, where our purpose is to *design* stable
dynamic systems.

As an example, we might be interested in examining aspects of the stability
of an exchange-rate system such as that recently advocated by sixteen
distinguished academic economists [14] -a sliding parity system (with widened
exchange-rate
margins).^{9} Is it not precisely a set of conditions such
as the Hicks conditions that would be involved? We might ask, first, what
would happen if, say, Britain (which we shall identify with country 1) adopted
a sliding parity system while a subset of other countries (2, . . .,
*j*) allowed their exchange rates to float, and the remaining countries,
*k*, . . ., *n*, kept their rates pegged to, say, the U.S. dollar
(the currency of country 0). Then, if we suppose that balances of payments
of countries whose rates float adjust instantaneously while the pound adjusts
slowly, the path of the pound over time would be

for which knowledge of the Hicks conditions would be directly relevant. Thus the particular form of the dynamic system adopted -which countries are left out and which are left in- would depend on which of the Hicks conditions are satisfied. The Hicks method does, therefore, have a role to play in dynamic aspects of the theory of economic policy.

The Hicks *conditions*, however, are not exactly what we need for the
theory of economic policy, because, as we shall see, they are incomplete
even in terms of Hicks' own method. In the experiments Hicks conducts to
derive his stability conditions he accords the numéraire -the standard
commodity- a special role. In this section we shall consider the precise
deficiencies in the statical information provided by the Hicks conditions.
This is best established by considering the comparative statics theorems
implied by the Hicks conditions. Consider the equilibrium system

where, again, the *B*_{i}'s are balances of payments,
the *p*'s are exchange rates, and *alpha* is a parameter.

Differentiation of (11) with respect to *alpha* yields

and the solutions for the exchange-rate changes are

Now consider an increase in demand for the currency of country *i* such
that *deltaB*_{i }/ *delta alpha *> 0;
while the excess demand for every other currency, at given exchange rates,
is unchanged (*deltaB*_{j } */ delta alpha
*= 0 for *i* *not= * *j*). Then, instead of (13),
we have simply

By the Hicks conditions of imperfect stability *Delta*_{ii
}/*Delta* < 0, so *deltaB*_{i
}/ *delta alpha *> 0 implies
*dp*_{i} / *d* *alpha * > 0.
Thus an increase in demand for the currency of the *i*th country raises
the price of that currency after adjustment of all other exchange rates has
been allowed for.

Similar implications follow from the conditions of perfect stability if we
hold various subsets of other exchange rates constant relative to the
numéraire. If, for example, the exchange rates of countries *k*,
. . ., *n* are held constant, we get, instead of (14), the equation

the inequality being an implication of one of the conditions of perfect stability.

How can an increase in demand occur in a closed system ? Clearly only at the expense of other commodities (currencies) in the system. Cournot's law (or Walras' law in the context of the Walrasian system) ensures that

where the summation, it should be emphasized, extends over all the commodities.
The interpretation of (14) is therefore that an increase in demand for (say)
pounds (the currency of country *i*) at the expense of dollars raises
the dollar price of the pound. Now if other exchange rates are held constant
relative to the dollar, the proposition holds, if the Hicksian perfect stability
conditions are satisfied, when the shift of demand is interpreted as being
from the dollar and all currencies whose exchange rates are kept fixed to
the dollar. Note, however, that the Hicks conditions do not give us the sign
of

so that we cannot specify, on the grounds of the Hicks conditions alone, whether a shift of demand from dollars to pounds raises or lowers the price of (say) the franc relative to the dollar.

But now we are in a position to see the narrow form of the mathematical
implications of the Hicks conditions. Consider a shift of demand from the
franc (currency *j*) to the pound (currency *i*). Then
*deltaB*_{s} / *delta alpha *= 0 for *s
not= i*, *j*, while *deltaB*_{i} / *delta
alpha *= *- deltaB*_{j} / *delta alpha
>*0 in view of (16); with no loss of generality we can make
*deltaB*_{i} / *delta alpha* = *-*
*deltaB*_{j} / *delta alpha* = 1. Substitution
in (13) then gives the change in the dollar price of the pound and the franc:

The Hicks conditions do not provide us with the sign of either (18) or (19), nor, by analogy to (17), should we expect them to. But, by analogy with (14), we should expect the difference

to be unambiguous in sign for any system in which units are chosen so that
each *p*_{s} = 1, initially. When demand shifts
from the franc to the pound, we should not expect to be able to predict the
sign of the change in the *dollar *price of the pound or franc, but
we should be able to determine, on the basis of the Hicks conditions, the
sign of the change in the *franc* price of the pound, the expression
given in (20). But the Hicks conditions are no help here, and this means
that Hicks has not developed the mathematical implications of extending his
method to the standard commodity.

The same information problem applies, a fortiori, when various subsets of
prices are held constant. An implication of the Hicks condition of perfect
stability is that a shift of demand onto pounds raises the price of the pound
even when various currencies remain pegged to the dollar; this amounts to
treating the dollar and the other currencies pegged to it as a
*composite* currency. By analogy the price of the pound should rise
when demand shifts from a currency other than the dollar, say, the franc,
while other currencies (for example, the mark) are pegged to the
*franc*.

Thus consider a shift of demand, at constant exchange rates, among three
currencies *i*, *j *and *k*, such that

and every other *deltaB*_{r} / *delta alpha
*= 0. Then, from (13), we have

Applying the restrictions that and setting

we can deduce the change in price of the pound relative to the mark and franc.

where *A* is the sum of the cofactors of the elements in the following
matrix:

The cofactor of, say, the element
*Delta*_{jk} can be related to the second cofactors
of *Delta* by Jacobi's ratio theorem,

so that *A */*Delta* can be written entirely as the sum of second
cofactors, and (24) can be rewritten

The inequality sign should hold if an increase in demand for one country's
currency occurs at the expense of *any* other country, one other currency
price remaining constant relative to that country. But the mathematical
information is not given to us by the Hicks conditions. The reason is that
the mathematical implications of the Hicks method have not been developed
with respect to the currency adopted as
numéraire.^{l0 }

When we do extend Hicks' method to make it " symmetrical " with respect to
the numéraire (appreciating, say, the pound relative to, say, the
*franc*, allowing various subsets of other currency markets to adjust),
we get, of course, a set of conditions that specifies the signs of terms
like those in (20) and (25). Along with the Hicks conditions, which can be
written

[The next term requires that the ratio of the denominator of the second ratio
and the sum of sixteen third minors be negative, and so on for successive
ratios. The last term in the conditions of (27) specifies that the sum of
the (*n*-l)th minors (*n*^{2} in number)
be negative. But the (*n*-l)th minors are equivalent to the elements
in the original determinant, so the last condition simply requires that the
sum of all the elements in the original determinant *Delta *be negative.]

This suggests an alternative-and simpler-way of developing the generalized conditions. Consider the augmented determinant

formed by bordering *Delta* with its column and row sums, with a change
of sign, so that

Then the extended Hicks conditions can be stated simply as the requirement
that principal minors of *B* arranged in successive order oscillate
in sign, except for the (singular) determinant *B*
itself.^{11}

Because the elements in the augmented determinant *B* are interdependent
the generalized conditions can be expressed entirely in terms of the elements
of the "normalized" determinant *Delta*. The supplemental conditions
are

with the last condition reducing to the basic determinant *Delta* itself.
These forms are equivalent to (30) and imply the signs of the ratios in (27).

This representation has the intuitive appeal of starting with the matrix
of *all* the currencies in the
system.^{l2} Thus, instead of omitting the
numéraire currency at the outset, we start with a nonnormalized system
of *n* + 1 currencies, exchange rates being expressed in terms of an
abstract unit of account (for example, IMF par values), and apply the Hicks
conditions allowing each currency the role of numéraire in turn.

The above conditions are more general than the Hicks conditions. Yet they
still do not exhaust the information inherent in the Hicks methodology. The
Hicks method of holding various subsets of prices constant with respect to
one another can be regarded as a device for constructing "composite commodities";
in the present context of currencies, we shall describe them as "currency
areas." Now if we apply the Hicks method to a system based on arbitrary
arrangements of countries in the currency areas, we get a further generalization
of the results obtained by Hicks. When, for example, the mark is pegged to
the dollar (the numéraire), the dollar and mark constitute a currency
area. But there is no reason to restrict the formation of currency areas
in a unidirectional attachment to the dollar. A group of currencies could
be "attached" to the pound or the franc or any other currency, in principle.
More importantly, we can then allow entire currency areas to appreciate and
require that the balance of payments of the areas worsen, while various subsets
of other currencies in the currency areas remain
unchanged.^{13}

The remarkable fact is that the conditions resulting from making arbitrary currency alignments among the (nondollar) countries and applying the Hicks method to the resulting matrix incorporate the conditions just developed as a special case. Thus consider the denominator of the first term in (27),

This term is the result of combining the *i*th and *j*th currencies
together to form a currency area of those two countries. With no loss of
generality we can write *i *= 1 and *j* = 2. Then if the first
and second rows and columns are replaced by their combined rows and columns,
we have

as can be proved by straightforward expansion. Similarly, it can be shown
that the denominator of the second term in (27) is the determinant formed
by replacing the *i*th, *j*th, and *k*th rows and columns
of *Delta* by the amalgamated row and column.

When we now carry out Hicks' method for arbitrary arrangements of currency
areas, extended over the whole range of currencies, we get a new set of
conditions on the original (*n* x *n*) price matrix. These conditions
can be expressed in a triangular arrangement of principal minors as follows:

The conditions on the left side of the stability triangle are the conditions
of perfect stability Hicks developed; they do not provide the information
implicit in extending the analysis to the numéraire commodity. The
conditions on the base of the triangle result from extending the Hicksian
method to the numéraire; they ignore the experiments resulting from
allowing currency areas to appreciate. Finally, the conditions on the right
side of the triangle are the conditions applicable when various sets of prices
are raised in the same proportion, other prices remaining constant. More
generally, the Hicks conditions on the left correspond to Hicksian adjustments
when each currency (commodity) is treated in isolation; the adjacent conditions
to their right are the Hicksian conditions when in the *i*th and
*i*th goods move in the same proportion; and so on. The entire set of
conditions are needed if the logic of Hicks' method is carried out to the
bitter
end.^{14 }

An important implication of the general conditions is that a system satisfying
the Hicks conditions, but not the general conditions, will be stable or unstable
depending on which currency is adopted as the key
currency.^{15} Consider, for example, a world of
three currencies, dollars (currency 0), pounds (currency 1), and francs (currency
2), and suppose that the balances of payments of the three countries are
related to exchange rates according to the equations

where the exchange rates are defined in, and the
*B*_{i} are expressed in, an abstract unit of
account (IMF par value units). (Equilibrium exchange rates are unity or any
multiple of unity, as the system is homogenous of degree 0.)

Let us consider a dynamic system in which the dollar is constant with respect
to its par value so that the dollar becomes the effective numéraire.
Let the par values of the pound and franc adjust in proportion to
*B*_{1} and
*B*_{2}, respectively. We then have the following
dynamic system:

In this system, the Hicks conditions of perfect stability, narrowly interpreted,
are satisfied (since *b*_{11} = -2 < 0,
*b*_{22} = -1 < 0), and

and the system is dynamically stable regardless of the (positive and finite)
values of *k*_{l }and
*k*_{2} .

Consider, however, a system in which the par value of the franc is fixed so that it becomes the "key currency " instead of the dollar. The dynamic system then becomes

for which the Hicks conditions are not satisfied, It is dynamically stable
or unstable according to whether *k*_{0}
>_{<} 4*k*_{l} .

This result could be predicted at once by applying the general conditions as given in the stability triangle (32). The sum of the coefficients in (34a) are positive, so the general conditions are not satisfied.

The general conditions are necessary conditions for a system to be stable
regardless of the currency (commodity) chosen as key currency (standard
commodity) and regardless of how quickly the various exchange rates adapt
to disequilibrium. This proposition is perfectly general in the sense that
it is valid in the *n*-currency
case.^{16}

I shall conclude this book by showing how the Hicks conditions, extended
as above, can be useful in devising dynamic mechanisms that are "strongly
stable." The problem could be approached from the direction of the correspondence
principle, which, in a narrow version of it, suggests that we apply to
comparative statics the conditions that the characteristic equation of the
systems have negative real parts. The justification for this narrow version
lies in the observation that the systems we know are not characterized by
instability. But there is no reason, in principle, why stronger conditions
could not be applied. We could, following Hicks (and Samuelson), require
that the system be stable no matter which subsets of market variables are
held constant. Alternatively, we could use conditions that the roots be real
or complex according to whether we observe cycles in the system under
investigation.^{17} If, for example, we observe an absence
of cycles in the real world, we know at once that the Hicks conditions are
sufficient conditions for dynamic stability.

In the theory of policy (under incomplete information) the problem is often
to choose among different dynamic systems, or different degrees of centralization
of a given dynamic system; this is often expressed in terms of allocating,
dynamically, instruments to targets (the problem of effective market
classification), and we may want to construct *strongly* stable systems:
first, because systems near the borderline of stability may become unstable
if disturbed by outside shocks; second, because the cost of adjustment may
be higher if the system, even though stable, oscillates in its approach to
equilibrium; third, because slight errors in manipulating rates of changes
in instrumental variables (interest rates, exchange rates, and so on) may
turn a weakly stable system into an unstable system; and, finally, because
the time involved in approaching equilibrium may be less under strongly stable
systems and rapid adjustment may be preferred to slow adjustment.

We can consider, therefore, the problem of choosing a dynamic control mechanism with "hyperstable" properties, in the sense that variables rise or fall whenever they are out of equilibrium, and show how the Hicks conditions can be of some help in constructing such a system when the precise location of an equilibrium is not known.

We take again as our example an international currency system. In a general hyperstable system it will be necessary to vary exchange rates, taking into account the balances of payments of each country. The problem is to find the weights each central bank should give their own balance of payments disequilibrium and that of the other countries.

Let *B*_{i} represent, as before, the balance
of payments of the *i*th country, dependent upon the *n* exchange
rates, according to the equation

that will make the system hyperstable. (The "speed"
*k*_{ij }can be interpreted as the weight that
country *i *has to give to the condition of the balance of payments
of the *j*th country in adjusting its own exchange rate.)

It is readily shown that the system (36) is hyperstable if the *k*'s
are chosen so that

where *alpha*_{i} is a negative real constant
and the *Delta*_{ji}'s are, as before, the cofactors
of *Delta*. The condition (37) means that the hyperstable speeds are
weighted elements of the inverse of the price matrix.

To prove this proposition we need to prove that the dynamic system

But, from the properties of any determinant, the typical term

has a value of unity for *k* = *i* and a value of zero for *k*
*not=* *i*. The system (39) therefore reduces to

It is instructive to write out the hyperstable system (38) in detail to see clearly the implications of Hicksian perfect stability for dynamics:

We shall also find it convenient to consider a reduced system in which we
choose the *alpha*_{i}, the *rate* at which
each *p*_{i} is restored to equilibrium (with
a negative sign), to be equal to the corresponding Hicksian conditions of
imperfect stability as given in (9), that is, we equate

Two observations can immediately be made about (44). First, if the elements
of the inverse matrix all have the same sign, hyperstability implies that
positive weights be assigned to each balance of payments. Thus, "Britain"
should depreciate (appreciate) more rapidly the greater the deficits (surpluses)
in the balance of payments of other countries, for any given deficit in her
own balance (this implies corresponding changes in the U.S. balance). But
from Mosak's theorem the elements in the inverse will all have the same sign
if the original currency matrix
[*b*_{ij}] is a gross substitute matrix provided
[*b*_{ij}] is Hicksian; and it will be Hicksian
provided the dollar is also (reciprocally) a substitute for all other currencies.

The second point to notice about (44) is that " Britain" should attach less
weight to the balance of payments of other countries' currencies than to
her own balance if currencies are all substitutes for one another; this follows
because every ratio *Delta*_{ji }/
*Delta*_{ii} < 1 for *j* *not=*
1.^{19}

Leaving now the special case of the gross substitutes to return to the more
general case represented by equations (42), we can find immediate implications
of the Hicks conditions. First, if the Hicks conditions are satisfied, "
normal" adjustments are implied in the sense that, *ceteris paribus*,
a deficit in a country's balance of payments suggests depreciation and a
surplus appreciation; this follows because every
*k*_{ii} =
*Delta*_{ii} /*Delta* > 0 given the Hicks
conditions of imperfect stability and *alpha*
_{i}< 0.

But more than this can be said. The following identities have to hold, from our definitions and Jacobi's ratio theorem:

the inequality following at once from the Hicks conditions of perfect stability. Proceeding to the last term we have

More generally, if the basic matrix
[*b*_{ij}] is Hicksian, the matrix of the speeds
required for hyperstability must satisfy the conditions that every principal
minor be positive, that is,

Analogous conditions hold for the extended Hicks conditions if the system is to be hyperstable regardless of the currency used as the key currency. In this sense the Hicks conditions alone are sufficient to establish in a weak sense the correctness of exchange rate policies directed at correcting "own" balances of payments.

These developments conform to the conclusions of economic intuition indeed,
they may be interpreted as bringing the mathematical treatment of the subject
closer to the level of common sense. They nevertheless suggest that the Hicksian
stability analysis is not lacking in significance for the integration of
dynamical and statical
theory.^{20,
2l}

______________________________

Notes

1 Adapted from: a forthcoming article in *Essays in Honor
of Sir John R. Hicks* (N. Wolfe, ed.).

2 I am happy to acknowledge many helpful conversations on the subject matter of this chapter with J. C. Weldon of McGill University.

3 Strictly, the problem is not trivial even in the case of
exchange of two commodities; first, because of complications associated with
the possibility of time derivatives of various orders of price changes entering
the excess demand (*X*) functions, in a system such as
second, because of nonlinearities in the excess demand function; and,
third, because market exchange of two commodities among many people may involve
adjustments of quantities toward budget constraints, giving rise to the more
complicated dynamics such as Marshall postulated in his foreign trade analysis.

4 Let
*X*_{i} =
*X*_{i}(*p*, . . .,
*p*_{n}) be the excess demand functions for
commodities *i* = 1, . . ., *n*. By differentiation
. Solving for the
*dp*_{i} we get
, where *Delta* is the
determinant of the system and
*Delta*_{ji}, its first cofactors. When all other
prices adapt so that every *dX*_{j} = 0 except
*dX*_{i}, we have the solutions
*dp*_{i }=
(*Delta*_{ii} /*Delta*)
*dX*_{i}; if various prices *k*, . . .,
*n*, are held constant with respect to the numéraire we get instead

Imperfect stability requires that
*Delta*_{ii} /*Delta* < 0 for *i*
= 1, . . ., *n*, whereas perfect stability requires that
*Delta*_{ii,kk,} ...
,_{nn} / *Delta*_{ kk}
... ,_{nn} < 0 for any subset of commodities
*k*, ...,*n*, excluding commodity i. The Hicksian perfect stability
conditions are thus

the first condition being the condition of imperfect stability.

5 It is slightly ironic that Marshall in his 1879 manuscript on foreign trade utilized the link between dynamics processes and stability, but not explicitly the link between stability and comparative statics, whereas Hicks utilized the link between stability and comparative statics but not the link between dynamics and stability. Samuelson used both links in his integration of dynamics and statics.

6 Actually, Metzler's example (p. 285) that the Hicks conditions
are not sufficient for stability to be independent of the speeds of adjustment
contains a numerical error that mars his demonstration [the second cubic
of his footnote 12 should read
*lambda*^{3} +
4*lambda*^{2} + 3.4*lambda* + 13.2 = 0 instead
of *lambda*^{3} +
4*lambda*^{2} + 2.6*lambda* + 13.2 = 0 and
in the first (correct) cubic the Routh conditions are satisfied]. But a slight
adjustment to his counterexample can nevertheless demonstrate his point.
If, for example, speeds are chosen so that, in his terminology,
*k*_{l} =
*k*^{3} = 1, giving the cubic
*lambda*^{3} + (2 +
*k*_{2})*lambda*^{2}
+ (1+ 1.2*k*_{2}) *lambda* +
6.6*k*_{2} = 0, the Routh conditions will not
be satisfied for values of *k*_{2} somewhat less
than 1.

7 In this connection Samuelson writes ([88], p. 554): "In principle the Hicks procedure is clearly wrong, although in some empirical cases it may be useful to make the hypothesis that the equilibrium is stable even without the 'equilibrating action' of some variable which may be arbitrarily held constant." Samuelson provides an example in connection with the Keynesian system later in his paper.

8 See, for example, Mundell [73] and the references there to the principle of effective market classification.

9 See page 111 for a list of economists who signed the petition.

10 This is perhaps apparent immediately, since the Hicks conditions can be expressed in terms of the commodity chosen as numéraire, the excess demand for which is never allowed to go to zero. On this point (but without apparent recognition of its implications) see Lange ([41], p. 92).

11 More compactly, we can state the conditions as requiring
that every first minor of *B* be "Hicksian." Not all the conditions
are independent, however, since *B*_{00}
__=__ *B*_{11} __=__
*B*_{n}_{n} in
view of the characteristics of *B*.

12 One might think that Hicks really intended his conditions
to extend over the entire range of excess demand coefficients (including
the numéraire), this would be incorrect since the last (augmented)
determinant is singular. Alternatively, it might be thought that Hicks intended
the conditions to apply no matter what commodity were taken as numéraire.
The latter interpretation seems to be fortified by a footnote which states
([24] p. 75): " [this] can be seen at once if we adopt the device of treating
*X* (momentarily) as the standard commodity, and therefore regarding
the increased demand for *X* as an increased supply of the old standard
commodity *m* ...." If this device were adopted to derive the Hicks
conditions it would indeed result in conditions equivalent to the above stability
conditions. Yet this interpretation would conflict, not only with all subsequent
interpretations of the Hicks conditions by other writers on stability, but
also with Hicks' own explicit statement in the Mathematical Appendix (p.
315), which specifies that the conditions must hold "for the market in every
*X*_{r }(*r* = 1, 2, 3, . . ., *n*-1)";
that is, the market for the numéraire (the *n*th commodity) is
omitted, In any case his discussion on pages 68-71 fails to make the point
clear, while his third graph in Figure 16, which he asserts is stable, is
actually totally stable only if the line along which there is zero excess
demand for the standard commodity (not drawn in his figures) is inelastic
referred to the abscissa. The discussion is not completely clear, however,
and I am therefore inclined to interpret Hicks as actually intending to give
full coverage to the numéraire but not completing the mathematical
implications of doing so.

13 I have considered the problem of " optimum" currency areas in terms of its attributes for stabilization policy, optimum exploitation of the functions of money, and so on in Chapter 12; the present analysis suggests an additional criterion in terms of stability conditions.

14 Stability conditions should, in principle, be "invariants"
in the sense that they are independent of the choice of units in which
commodities are measured; this corresponds to Lange's "principle of invariance"
([41], p. 103). The above stability conditions are invariants only if, as
is assumed in the case of the foreign exchange market analysis, the
*b*_{ij}'s are all measured in equivalent currency
units.

In the general case, however, where prices respond to excess demands denominated
in physical quantity units, the units in which
*b*_{ij}( *j *= 1, . . ., *n*) is measured
differ from the units in which
*b*_{kj}(*j* = 1, . . ., *n*) is measured.
This means that invariant stability conditions require that each row of every
stability determinant involving the sum of elements be multiplied by an arbitrary
number reflecting units of measurement. Thus "unit-invariant" stability in
the general case involves the sign of terms such as

For arbitrary values of *k*_{i }and
*k*_{j} extended over the entire range of the
(*n*+ 1) x (*n*+ 1)-*B* determinant, and given the homogeneity
postulate, the general conditions then imply that all goods are gross
substitutes.

15 From a historical point of view, it is somewhat amusing that although the Hicks conditions are not symmetrical with respect to the commodity chosen as numéraire, neither are the (normalized) dynamic systems used to refute the validity of the Hicks conditions as true dynamic stability conditions.

16 The proof in the general case follows by analogy to the proof of Metzler ([62], pp. 280-285) when his method is extended to include the augmented system.

17 I have applied this method to the problem of disentangling lags in expectational and cash balance adjustments [77].

18 More directly the matrix
equationif *k* =
*B*^{-1}.

The usefulness of this result lies not so much in providing a policy maker with the rules of adjustment, since there is no problem if the basic matrix is known, and the rule gives insufficient information if the basic matrix is not known. The point is rather that pieces of information about the inverse may be sufficient to distract policy makers from mistakes, and that conditions like the Hicks conditions may be a sufficient guide to the relative importance to be attached to particular instruments.

19 I have discussed the implications of this condition in some detail [78].

20 Even the gap between the true dynamic stability and Hicksian
stability may be bridged by integrating Hicksian stability conditions with
the "Samuelson-Le Châtelier principle" (see, for example, Samuelson
[93]) which can be expressed entirely in terms of the Hicksian determinants.
Hicksian stability and dynamic stability mesh together under appropriate
conditions of gross substitutes, gross complements, and symmetry, a common
link being sign symmetry; and sign symmetry of the inverse of the basic Hicksian
matrix is sufficient for at least one of the Le
Châtelier conditions to hold
(*DeltaDelta*_{ii,jj} -
*Delta*_{ii}*Delta*_{jj}
< 0 if *Delta*_{ij} and
*Delta*_{ji}, have the same sign).

21 A further implication of the Hicks conditions was called
to my attention by Daniel McFadden, who had proved, in a paper presented
at the 1963 Econometrica Society meetings in Boston, Massachusetts, that
if the Hicks conditions of perfect stability are satisfied, a stable dynamic
system of the form
can
always be found for a diagonal *K* matrix with positive diagonal
coefficients; the result also holds in the global form. Prior to McFadden's
result, and unknown to him, a local version of the theorem had been published;
see Fisher and Fuller [15].

In the context of the currency problem discussed above, the theorem means that if the Hicks conditions are satisfied, it is always possible to find a stable dynamic system in which exchange rates are adjusted to " own " balances of payments only.

____________________________________

[14] W. J. FELLNER *et al., Maintaining and Restoring Balance in International
Payments. *Princeton, N.J.: Princeton University Press, 1966.

[15] M. E. FISHER and A. T. FULLER, " On the Stabilization of Matrices and
the Convergence of Linear Iterative Processes," *Proc. Cambridge Phil.
Soc., *54 (1958).

[24] J. R. HICKS, *Value and Capital. *2nd ed. Fair Lawn, N.J.: Oxford
University Press, 1946.

[41] O. LANGE, *Price Flexibility and Full Employment. *Bloomington:
Indiana University Press, 1944, p. 92.

[62] L. A. METZLER, "Stability of Multiple Markets: The Hicks Conditions,"
*Econometrica, *13, 277-292 (Oct. 1945).

[73] R. A. MUNDELL, "The Appropriate Use of Monetary and Fiscal Policy for
Internal and External Stability," *IMF Staff Papers, 9, *70-79 (March
1962).

[77] R. A. MUNDELL, "Growth, Stability and Inflationary Finance," *Jour.
Pol. Econ.. *73. 97-100 (April 1965).

[78] R. A. MUNDELL, "The Significance of the Homogeneity Postulate for the
Laws of Comparative Statics," *Econometrica, *33, 349-356 (April 1965).

[88] P. A. SAMUELSON, "The Stability of Equilibrium: Comparative Statics
and Dynamics," *Econometrica, *9, 97-120 (April 1941).

[89] P. A. SAMUELSON, " The Stability of Equilibrium: Linear and Non Linear
Systems," *Econometrica, *10, 1 (April 1942).

[93] P. A. SAMUELSON, " An Extension of the Le Châtelier Principle,"
*Econometrica, 28, *368-379 (April 1960).

© Copyright Robert A. Mundell, 1968